On the regularity of the solutions to the Navier-Stokes equations via the gradient of one velocity component

We improve a regularity criterion for the solutions to the Navier-Stokes equations in the full three-dimensional space involving the gradient of one velocity component. Revising the method used in Pokorny and Zhou (2009, 2010), we show that a weak solution u is regular on (0, T) provided that del u(3) is an element of L-t(0, T;L-s), where 2/t + 3/s = 19/10 for s is an element of [30/19, 10/3] and 2/t + 3/s = 7/4 + 1/(2s) for s is an element of [10/3, infinity]. It improves the known results for s is an element of [30/19, 150/77) and s is an element of (10/3, infinity]. (C) 2014 Elsevier Ltd. All rights reserved.